theorem Th23:
  r <> 0 implies Carrier (r(*)LR) = r * Carrier LR
  proof
    assume A1: r<>0;
    thus Carrier(r(*)LR)c=r*Carrier LR by Th22;
    let x be object;
    assume x in r*Carrier LR;
    then consider v be Element of R such that
    A2: x=r*v and
    A3: v in Carrier LR;
    r"*(r*v) = (r"*r)*v by RLVECT_1:def 7
            .= 1*v by A1,XCMPLX_0:def 7
            .= v by RLVECT_1:def 8;
    then A4: LR.v=(r(*)LR).x by A1,A2,Def2;
    LR.v<>0 by A3,RLVECT_2:19;
    hence thesis by A2,A4;
  end;
